A particle is confined in a potential well such that its allowed energies are $E^n = n^2\epsilon$, where $n = 1, 2, \dots$ is an integer and $\epsilon$ a positive constant. The corresponding energy eigenstates are $\lvert1\rangle, \lvert2\rangle, \dots , \lvert n\rangle, \dots$ At t = 0 the particle is in the state:
$\lvert\psi(0)\rangle = 0.2\lvert1\rangle + 0.3\lvert2\rangle + 0.4\lvert3\rangle + 0.843\lvert4\rangle$.
What is the probability if energy is measured at $t=0$ of finding a number smaller than $6\sigma$?
Am I right in saying this would just be the sum of states $n = 1$ and $n = 2$ which is $0.5$?
Then I'm wondering how you would calculate the mean value and rms deviation of the energy of the particle in the state $\lvert\psi(0)\rangle$
How do I find the state vector $\lvert\psi\rangle$ at any time $t$? And therefore do the results calculated above remain valid for any arbitrary time?
The last thing I'm stuck on is, lets say the energy is measured and it is said to be $16\epsilon$. After this measurement, what is the state of the system, and what result would you get if you tried to measure energy again?